They also use film caps, because of two reasons:
1. Speed of discharge. They have low enough ESR to be useful.
2. Reversal. The closed circuit has resistance
and inductance, and it completes one or two cycles before the energy is used up. Electrolytics can't tolerate this (it's destructive).
If enough ESR is added, the inductance can be dampened so that no reversal occurs. This is fairly typical of electrolytics
anyway, but that still doesn't mean it's safe to use them: they aren't usually made to withstand the intense forces (both the mechanical force of surge current, and the stress of rapid thermal heating) of short circuit discharge. Not to say they can't be, but that's added cost to make custom ones that can; and you might be better off using commercial film capacitors by then.
The minimum energy required to fire a railgun is apparently some kJ. It's not a problem that scales down well. You must have enough current flow (I >> 10kA) to create a strong enough magnetic field (H >> 10^5 A/m) to have enough pressure (p = mu_0 * H^2 / 2) to overcome friction, let alone to be anywhere near worthwhile.
The main thing you save is voltage (EMF, ultimately dH/dt), because the rail is shorter. But that's its own difficulty, because the impedance is that much lower.
You can see already that the Navy's railgun impedance is very low indeed. Those cables: they aren't simply solid copper. They are
kickless cables, where the positive and negative are braided together -- like Litz cable, but both polarities intermingled (still fully insulated), for the least possible stray inductance. And they have a lot of inductance to deal with, since that barrel will only be a few uH at full length, and they've got maybe a hundred feet between the gun and the furthest shipping container-load of capacitors. That would be a lot of stray inductance otherwise. (Plus, as the name suggests, the "kick" is confined inside the cable -- all that force is distributed over many strands, hence why the sets of cables don't explode away from each other like the shot itself does. Yeah, they flop around still, but that's just the rail's recoil.)
And, on top of that, they've got tons of cables wired in parallel. The total transmission line impedance will be fractional ohms, maybe 10s of milliohms even. If that thing's charged to a modest, say, 1000V, the surge current can be hundreds of kiloamperes, maybe a megampere or more.
Which, if the inductance is on the order of single digit uH, the capacitance would be on the order of whole farads. (This comes from the relation: Vpk / Ipk = sqrt(L / C).) Hmm, hard to say: that's a truly preposterous amount of film capacitors, if so. But if it's 1000V, you'd need 120F to store the claimed 60MJ, so it's probably that the voltage is higher, and the amount, well, it's still preposterous...
Tim