Federation computers in TNG seem to use some semiconductor/metal based substrate for computation. In the episode Evolution where nanobots are let loose by the boy wonder, they start to consume the computational core of the computer, and this rules out exotic substances or even organic ones. It is unclear how the Federation has subverted the end of Moore's law if this is the case however (which will happen to us within at most, decades).
It is possible that they utilize some quantum computational trick, which may actually be possible with mundane materials. Or it may be that running a starship such as theirs is possible with computers not so distant from our own capability level. It's difficult to speculate.
We do know that they are binary in nature, from several different episodes (the Bynars use binary, and no mention is made of the Enterprise using something profoundly different). They aren't using the 8 bit word (byte), or any obvious multiple of it. This itself isn't all that shocking, even in the United States in the last few decades we've had machines that used all sorts of word-lengths... DEC (Digital Equipment Corporation) was doing so well into the 1970s.
It's difficult to say with certainty, but the TNG computers probably do not use the Von Neumann architecture (which is what our CPUs use currently, almost nothing else). They may use the Harvard architecture (used commercially by only a few microcontrollers I can think of) or some variation. I say this because they seem absolutely naive about many different computer security implications that are a real problem today, and I suspect this is because those sorts of hacking attacks just aren't possible with their machines. I do not believe these are solvable entirely in software, either, though that would be contentious from a compsci point of view.
Also, we can tell from their database terminology that it is almost certainly not based on RDBMS concepts that we have today. They have way too much encyclopedic information at their fingertips, and use terms like "data stores" and the like.
Storage-wise, they're almost certainly able to do a single-bit-per particle/atom, and maybe even higher densities. The Enterprise has capacities that one must imagine to be in the exabytes or higher, and without the warehouse-sized server rooms that would require today.
The "neural gel packs" and so forth in Voyager were the result of bad writing, someone was on a search for new technobabble that week and pulled something plausible-sounding out of a Popular Science article. It's implausible that they would be experimenting with that at that point in the time, it should have happened centuries earlier in Star Trek' timeline or not at all, and it's also implausible that they'd be facing some sort of Moore's-Law-esque looming obstacle after having taken more mundane techologies so far.
