Can I build a linux distro with libc instead of glibc

Context: assuming from above comments that a BSDish libc is meant.

I think it's been looked into, but libc tends to be tightly tied to a given kernel (glibc has an abstraction layer, which allows it some portability but causes the usual problems that an abstraction layer causes) and making BSD libc work with a Linux kernel would require a near complete rewrite. key system services are very different between the two systems (one example: BSD libc assumes that there are no pipes/FIFOs, because BSD uses socketpairs instead; conversely, Linux doesn't support pipe-compatible socketpairs).

Going the other direction (Debian has an experimental Linux userspace on a FreeBSD kernel, I think) is possible due to glibc's portability layer.

There are many libc implementations. It is possible and quite commmon to build a system using uClibc or EGLIBC. It's not even particularly difficult: try Buildroot.