looking at building a PC or Mac system to handle its large files.
If you're happy assembling a PC from components, that's definitely the best way to go for value for money.
To get the process rolling;
First budget for a good monitor; Without a good monitor you're wasting your time. NEC wide gamut is generally well regarded here as the best option.
Buy a fast i7 processor, min 16gb ram and fit one, preferably two, SSDs (one for the OS, one for scratch disk/LR library). A fast USB3 card reader will also be a useful cheap addition.
You don't need any particularly fancy high end graphics card.
Back ups, as others have pointed out, are important, but I think that should be regarded as 'running costs' rather than part of the initial capital spend. Once you've got the new system up and running you can reuse any old drives in enclosures as external back ups. Only buying new external drives when needed.