You've got a range of choices, but you're looking at the $500 to $1000 price range for the mixer/interface. eg. Behringer XR-12/16, Presonus 16R, Soundcraft UI16 and the like.
You'll need half-decent computer with lots of storage (external 1TB SSD is a good option) AND some recording software (DAW or similar). The current USB. 4.0 spec handles this amount of audio bandwidth just fine.
The Xr-12 and xr-16 can't do multitrack recording, only stereo mixdown to a thumb drive. I just bought one used here on talkbass for solo shows. The Xr-18 is great for this -- I helped a friend who runs sound record a band's live show to my laptop. It got all the individual tracks without a single glitch for over two hours of music, and they mixed it and made a CD of it.
Of course you're committing to using a digital mixer for your rehearsals, which has a pretty big learning curve.
The AW4416 that somebody mentioned earlier is great for recording, but pretty wonky to set up as a live mixer in my opinion. They're quite cheap used, which hurts cause mine cost me near $3000 back in the early 2000's. The audio quality isn't nearly as good as my Focusrite Scarlett that cost me $75.