though I don't fully understand at what size number these modules become necessary
*Much* smaller numbers than the ones you're dealing with. The details are platform-dependent; for instance, a 64-bit platform can handle, without a big-number library, larger numbers than a 32-bit platform. But nobody is ever going to manufacture a computer that can natively handle numbers of the size you're talking about. Based on Moore's law, you couldn't expect such a computer to be available while Earth is still inhabitable.
You may want to look into ways to approximate factorials. Irrespective of what number library you use, I'm not sure it's possible to calculate the exact value of the factorial of 700 in reasonable time on a household computer.
Alternatively, look for ways to shortcut the problem, by simplifying before you multiply. You may find that some things cancel out in ways that save a lot of time. Also, if this is for a math class, you may want to check, as some math teachers are quite happy to receive answers in expression form, with not all of the calculations performed, because they're more interested in whether you understand how to work the problem than in whether you can multiply together a few hundred numbers.
"In adjectives, with the addition of inflectional endings, a changeable long vowel (Qamets or Tsere) in an open, propretonic syllable will reduce to Vocal Shewa. This type of change occurs when the open, pretonic syllable of the masculine singular adjective becomes propretonic with the addition of inflectional endings." — Pratico & Van Pelt, BBHG, p68