While you see zero mass fields often enough in Scifi to be believable to a general audience, the Higgs Mechanism pretty much disproves them as a possibility.
The Higgs Mechanism is the prevailing theory right now about where mass comes from. It explains that mass is a function of the interactions between fundamental particles. For Electrons, it is caused by drag with the Higgs Boson Field, and for Protons and Neutrons, it is mostly caused by the much stronger interactions between Quarks and Gluons. So to eliminate Mass you would need to eliminate all of the interactions between all of the fundamental particles in your spaceship. This means there would be no forces left to hold you together... at least until the field generator comes apart and all of your matter has to find a way to reassemble.
Best case scenario this will turn you into a sort of nuclear bomb. Worst case scenario, it will turn you into something much worse: a Quark Bomb. Quarks are bound together by the strong force which makes them really really hard to get apart, but if you eliminate the strong force for a while, all the quarks in your body will drift apart... but when your anti-mass field disappears, they will snap together with the strongest force known to man. Quark bombs are a lot of theory with only a tiny amount of observable data, but various models indicate that a quark explosion could be anywhere from 8 times as energetic as a same mass hydrogen bomb reaction to being even more powerful than a matter/anti-matter reaction.
So let's say you have a spaceship the size of the Enterprise-D, and put it into an anti-mass field. The resulting Quark explosion would be anywhere from about 6 to 107 billion megatons assuming you cap your explosion as a pure matter to energy conversion.
But as I said, a quark bomb might be even more powerful than antimatter. This may seem to violate the conservation of mass and energy, but the strong force becomes stronger as you move two bound quarks apart instead of weaker like electromagnetism or gravity. I don't know exactly how scientific this is, but some authors have taken this to mean that a single neutron can store infinite potential energy based on how far apart you pull its bound quarks. So, it is possible that your ship could even explode with all of the force of a supernova wiping out all life in the solar system.
So, zero mass fields make for GREAT weapons and possibly some very interesting power reactors, but not so much good as a form of propulsion as described.
Instead Use a Near Zero Mass Field
One of the problems with math & physics is that it tends to fall apart when you start working with zero or infinite values. So, instead of a zero mass field, what happens if you just reduce your mass? The attraction between your fundamental particles will become weaker, but so will their inertia so an atom with 1 millionth of its normal mass will only be held together by 1 millionth of its normal binding forces, but it is also 1 million times as easy to manipulate; so, it could still (in theory at least) hold its normal shape and form acting totally normal within your local physics system.
So lets say your ship is flying along at a humble 10 km/s, but you want to be moving at 0.9 c, you just reduce your mass to about .00000014% of normal. You are WAY less massive and have way less inertia, but still have a proportional amount of binding energy to inertia to maintain normal cohesion.
This might also help protect you from relativistic collision problems. Inside your own field you are still only experiencing forces comparable to moving at 10kps; so, even if you run into some space dust at 270,000 km/s, the dust will become part of your local physics system and the impact will feel like a 10 km/s impact: which you can much more easily armor you ship against.
Local physics fields of any sort are still pretty soft-science any way you add it up, but at least this should pass some basic believability checks as to why you don't just explode.
... and maybe use a Quark reactor to Power your Ship
If you have the ability to just turn on and off the fundamental forces of the universe, then depending on what model you go with, you may also have a virtually infinite power supply. Your ship could have a reactor that simply takes a tiny amount of regular matter, lets it drift apart far enough, and then slam back together releasing more energy than your fuel has mass. Since your ship can now violate the conservation of mass and energy, you could in theory accelerate your ship past the speed of light. Or if you go with a model that says a Quark explosion maxes out as a 1:1 mass to energy conversion, then you still have something as efficient as an antimatter reactor, but much easier to control and use for practical purposes.