The Google quiche repository is set up to be embedded in another project (e.g. chromium or envoy) and be built by that project. You can build using bazel (
https://bazel.build/) - from the quiche repository, you can run "bazel build //quiche:quic_client" to build the quic client. However, the bazel build files are incomplete - there's no target to build for quic_server. You'll need to provide your own implementation of parts of the quiche platform api, e.g. in a clean checkout quic_client fails because there's no proof verifier provided. I'd suggest looking at chromium or envoy's use of quiche to find implementations that might work.
117GB sounds too large for a depot_tools checkout - on my machine it's less than 1GB. My chromium checkout is 72GB, so I'm assuming you're looking for ways to avoid needing all of chromium.