"It is not possible". Or perhaps is it??
(1) How enemy might detect us
(1.1) Active. This is the active way of detecting. Enemy will emit some signal. This signal will interact with the hull. Iff hull reflects, then the enemy will pick up the signal. Analysis of the signal will identify our position, our speed, and possibly more data about us.
(1.2) Passive. This is a passive way of detecting. Enemy will not emit a signal, at all. Enemy will just listen for signals. If we emit a signal, enemy will detect us. If we reflect from signal emitted by someone else (say.. sun.. or another ship), enemy will detect and gather data about us.
(1.3) Cast Shadow. Other sources emit signals continuously at all directions. Say.. The sun. If the enemy is looking at sun, and their sensors detect the sun stopped transmitting its light in some small dot! Since the source emit continuously by hypothesis, conclusion is something is in between the sun and enemy ship. Something has been detected. Detection of cast shadow might be done with light of stars, sun, continuous radio emissions, etc.
(2) Detection by EM Spectrum
(2.1) Radio and microwave. RADAR is a system that actively scans regions of space, by emitting electromagnetic radiation mostly in microwave range, and hoping they will hit our ship and reflect back. Also, they might not emit any radiation at all, instead, they will listen for any emissions we do. If we try to contact someone using radio, we give away our position. Other radio sources that is emitting radiation might hit us, reflect from us, and enemy might listen.
(2.2) Passive infrared detection. The enemy will try to look infrared signals. They are often released by black body radiation in normal temperatures. Basically, a thermal detectable signature because we have non-zero temperature. Also, other infrared sources might hit us, reflect and go to enemy, making us detectable.
(2.3) Passive visible light detection. Looking.. maybe with naked eye. Or telescopes. Or whatever. We might be emitting visible light (lamps, engines, or whatever). Or, any other visible light sources (say.. sun, etc) might be emitting, hitting us, and reflecting back to enemy, which in turn it will detect.
(2.4) Detection by radiation of higher energies. Cosmic radiation interacting with matter might emit photons, which can be detectable (to a ultra amazing high ridiculously accurate detector). Maybe we should ignore this situation. Or postpone to a future version.
(3) Effective cloak.
There are 4 possibilities of interaction of classical light and us: Reflection, Transmission, Absorption. In addition, there is the possibility we are emitting such things.
(3.1) Emission. We cannot emit any of these signals (radio, microwave, infrared, visible, higher energies). If we do, they will detect us. Its easy to avoid emitting radio, microwave. Infrared is tricky. Visible is easier. And higher energies is special.
(3.2) Reflection. We cannot reflect any of these signals. Its easy to avoid reflecting radio, microwave. Somewhat hard with infrared and visible.
(3.3) Transmission. This is the ideal. We must transmit everything.
(3.4) Absorption. This is good, but not good enough. With careful looking and amazing algorithms, the enemy ship might detect that some stars are disappearing and appearing back (because we are absorbing their light). The same apply with natural radio emission sources, infrared sources, maybe even X-Ray sources).
(4) Microwave and radio cloak
(4.1) Emission. Trivial to avoid.
(4.2) Reflection. It is not that hard to prevent reflection of microwave and radio sources. The reason for this, its because their wavelengths are sizes we are costumed to (from meters to centimeters). By choosing the right geometry in your ship, to reduce RADAR cross section at maximum you can, you will avoid reflection.
(4.3) Transmission. This is ideal, as we said. Thus we need to transmit the maximum we can. But it is somewhat hard to transmit radio signal, precisely because their wavelength is big. There are few ways of doing this.
(4.3.1) Making a ship transparent to this kind of radiation. This means, in all practical purposes, making a ship made fully of dielectric. Metallic or electrically conducting surfaces reflects this kinds of signals. Hardly feasible. And even pose some problems: By Snell law, the transmitted signal will shift angle and thus if enemy is far enough, we will be effectively blocking the source. Conclusion: (22.214.171.124) Offers good protection against active radar. (126.96.36.199) Offers good protection against natural sources. (188.8.131.52) Offers bad protection against shadow cast. (184.108.40.206) Unfeasible.
(4.3.2) Redirection. We can redirect all incoming radio signals, such that their Poynting vector remains unchanged from what it was before hitting us. (220.127.116.11) Using wave-guides, this is extremely hard precisely because their wavelengths is too big. A wave-guide that guides EM Radiation of wavelength $\lambda$ needs to have a size of approximately $\lambda$. Your ship would have to be huge and covered with wave-guides. (18.104.22.168) Using nice engineered materials (say.. meta-materials), this task becomes somewhat "simple" precisely because the wavelength is big. Some methods have already been developed
(4.4) Absorption. Since transmission (the ideal one) is unfeasible, we shall place our hopes here. And thank for us, absorption is possible.
(4.4.1) Geometry. We can make the ship in such geometry to reduce active radar cross section at maximum we can. Then the signal of enemy radar will mostly be absorbed and be made undetectable.
(4.4.2) Material. There are absorbing materials that we can place in the hull, such that it will absorb incoming radiation. There also exists engineered materials (like meta-materials) made to absorb radiation in this range of frequencies.
(5) Infrared cloak
(5.1) Emission. This is terribly hard to avoid. Our ship emit infrared radiation because of black body radiation, which mostly fall into the infrared, using Wien's law. ) According to Wien's displacement law: $\lambda = b/T$, where $b$ is Wien's constant. If we decrease the temperature to shift from infrared to microwave, your temperature would need to be unfeasible low. In addition, you must radiate waste heat, if you plan the ship inhabitants/computers/equipments to survive. To effectively cloak our selves, we need to make sure enemy does not receive our black body radiation.
(5.2) Reflection. Hard Hard Hard. We cannot reflect from other infrared emissions if we plan to be undetected.
(5.3) Absorption. If we absorb, we must be careful with the shadow cast problem, so you cannot be in between infrared sources and your target. This is very hard in star systems. And hard in open space (since stars of course do emit infrared, and they are everywhere). Absorption only is dangerous and risky. If we plan a good cloak, we cannot absorb. There is only one left: We must transmit.
(5.4) Transmission. Here we are. Its a must. If we transmit, we avoid active and passive detection. Our biggest problem is cast shadow: There is continuously emitting infrared source $S$, and enemy ship. We are in the middle. If the transmission is perfect, light from source will pass thru us if we didn't existed. If there is a delay, may be detectable if we are moving at an certain speed or higher. If it is not perfect (likely), there might be distortions in the transmitted light, and aberrations. If enemy calibrate their scans, they might detect this distortions/aberrations, and move to investigate (or perhaps to simply shoot at it, just in case).
(6) Visible Light.
(6.1) Emission. Trivial to avoid. Just shut down all the lamps in the outer-hull.
(6.2) Reflection. We cannot reflect from other visible light emissions if we plan to be undetected
(6.3) Absorption. Same argument for infrared. If we plan a good cloack, we cannot absorb visible light.
(6.4) Transmission. Here we are. Its a must. And same argument of infrared applies to visible light.
(7) What we need.
(7.1) Geometry. Making things in the right geometry to reduce radar cross section and be invisible by radio and microwave. Maybe a fractal pattern. Or something to make the waves cancel out. I don't know.
(7.2) Infrared and Visible light transmission. We need to perfectly redirect the infrared and visible sources as if we are not there. This is proving to be quite difficult with our current technology level.
(7.3) Black body Radiation. Following (5.1), we need to make sure enemy does not receive our black body radiation. If you have hypothetical material that nicely does (7.2), then it will not heat up (since there is no conduction, no convection, only heating by irradiation in space, and (7.2) transmits everything, not heating it up). After some time, it will be as cold as environment. Since at principle you know where the enemy is, you can transform inside-ship black body spectrum into unidirectional beam, and waste it away not pointing the beam at enemy. That way, you remain undetectable, and you have successfully dissipated your waste heat.
(8.1) Keeping people inside alive. (7.3) solves the waste heat. Life support system is not disrupted by any of this. Then, you can keep people inside alive.
(8.2) Carry out the normal duties. Yes. There is no reason why not. As long as the ship is shielded internally (so radio signals from inside does not reach outside). And shielded infraredly and visibly (easy). Be careful as to your power source. It may release some kind of detectable radiation that might make its way to the outside ship (maybe neutrons, or gamma rays).
(8.3) Keeping your sensors on. Of course!! You will only need to shutdown your active sensors (for obvious reasons). And leave only the passive sensors online (which is more than enough to detect an enemy ship). Enemy thermal signature alone might do the trick. And if enemy ship has active radar online, you can detect its position by the incoming radio signals. =).
(8.4) Still able to maneuver. Tricky. Your engines cannot release any radiation at all. The only only way I can think off to accomplish this, is a special propulsion to operate in cloak only, which shoots projectiles. Very cold projectiles (temperature of background universe), which does not reflects radio, microwave, infrared, visible. Hard, isn't it? Perhaps you could engineer such projectiles to be such way: To cover the projectile with the same hypothetical material that keeps your ship cloaked in (7.2).
Let me know where this answer can be improved. =).