In terms of our current computers, you're probably looking at something like yottabytes .. but they don't seem to store things in bytes at this point, as others say, quads. Which could mean that they've knocked the basic storage unit size from 'byte' to '4 bytes' ... A current 64-bit system can address 16 exabytes of memory, but if you were to knock that upward by an exponent of 4, you could address 65k exabytes, or 0.06 yottabytes.

That's if we're talking about memory capacities. Warm storage capability / use tends to be significantly larger than memory use, so I'd hazard a guess that the total storage capacity of a starship's computers might be in the area of a few yottabytes.

This is assuming that "quad" is used because more space was required for memory architecture, and rather than upgrade to 128-bit or 256-bit computers, necessitating a huge amount of code refactoring, they just increased the minimum storage length. Of course, that would also realistically require a huge amount of code refactoring, but we're all just making stuff up here, right? :D