You envision these ships "sitting" in space, but nothing truly sits in space. Everything is constantly moving along curved paths through gravitational fields. In order to not consume obscene amounts of fuel, these ships need to find a location that will let them naturally move in such a way that they are always between the Sun and the Earth. Fortunately for your attack fleet, such a point exists: it's called the Sun-Earth Lagrange point 1, or SEL-1. Objects placed at this point will require very little station-keeping to stay balanced between the Earth and the Sun. This point is outside of the Moon's orbit, approximately 1% of the distance from the Earth to the Sun. The wiki says that we currently have five active probes in the vicinity of SEL-1, including the solar observatory SOHO, and the climate observatory, DISCOVR, which took the following photograph:
Size of Shield
Before we can determine the number of ships, we need to know the total size of the shield they need to form. Imagine a very long cone. At the large end is a giant ball for the Sun. Near the small end is a smaller marble for the Earth. Even though there is a huge size discrepancy between the two, the length of the cone (the distance from the earth to the sun) is so much larger than the Sun, that the cone is still relatively narrow (hopefully that picture makes sense).
In order to block all the sunlight, the shield needs to be a disc wide enough to fill the cone at its location. As a reminder, this disc is located 1% of the way from the Earth to the Sun. This means that the disc of the shield needs to be larger than the Earth, but not by very much (because the cone is narrow, and the shield is relatively close to earth). In fact, rather than bother with the exact trigonometry, I'll just do a rough estimate and assume the shield has approximately the same radius as the Earth.
This leads to an area of:
Ashield = π * Rearth2 ~ 128,000,000 km2
Size and Number of Ships
You want a ship "constructable with current physics" which I assume means something roughly on par with today's engineering capabilities.
Solar sail technology is still in its infancy, but what was originally planned to be the world's largest solar sail (granted, not a hard bar to surpass), was the Sunjammer. Interestingly, before it got cancelled, it was meant to launch in 2015 with DISCOVR towards SEL-1. It was to have an area of 1200 m2 (which is only 0.0012 km2).
The ISS is the largest man-made object in space, and its solar panels have an area of about twice that, at 2500 m2 (0.0025 km2).
Let's stretch the limits of what is currently possible, and assume for the sake of simplicity that, with some effort, we are able scale this up by about 3 orders of magnitude, and build and deploy a large solar sail with an area of 1 km2 (for comparison, the Golden Gate bridge is 2.7 km long). Then, given the required area we computed above, we would only need 128 million of them.
Given enough lead time, we can get an object pretty much anywhere in the solar system... or in some cases, even outside of it. The length of time that it takes to get to a destination is very much determined by the trajectory taken and the type of propulsion used. DISCOVR took 100 days to reach SEL-1, but it was no doubt taking a fuel-efficient route. If we took a more energetic route, we could probably get an object there within a week or so (just spit-balling it, based on a 3-day lunar trajectory used for Apollo, and a need for urgency)
However, we do not currently have the weaponry to take out a literally-planet-sized flotilla of ships. Or do we? According to this list, it looks like if we pooled our international resources we could muster about 4,000 active nuclear weapons. Mark makes an excellent point about the damage of these weapons being minimal in space, since there is no gaseous medium to transmit a shock wave. Even if we hit 4,000 of the 128 million sails, that's less than 1 percent of 1 percent of the total fleet. But... maybe we might get lucky, and initiate a Kessler Syndrome-style chain reaction. After all, these sails are presumably flying in a rather tight formation in order to block the sun. Taking out one might be enough to take out its neighbors, which each take out their neighbors, and so on. Whether this actually happens would depend on a number of factors, including the toughness of the sails, the tightness of the formation they are flying, and the force of the nuclear weapons detonated.
Even still, it's not clear how helpful this would be, because you've still got a massive debris field sitting between Earth and the sun. Without active station keeping, the debris will eventually start to dissipate, but this can be expected to take on the order of at least several months. Probably longer.
So in short, our defenses would consist of building enough rockets to launch every single nuke on the planet to SEL-1, waiting at least a week for them to get there, then hoping that they can initiate a chain reaction that leads to a debris field that will eventually disperse after several months. I hope someone can come up with a better defense plan than that, because I don't think that would be sufficient.
On the bright side, if each of these sun-blocking ships only cost the projected amount of Sunjammer (\$20 million to build + \$10 mil. to launch) -- despite having 3 orders of magnitude more surface area -- then the total cost of the shield would be \$30 million per ship * 128 million ships ~ $3.8 quadrillion. That's about 200 times the US national debt, so we're probably safe for now.