Yes, and there are certainly cache-timing attacks between processes (and even VMs) on the same physical hardware. That said, if you're disciplined enough to avoid all its awfulness, C is one of the least-bad languages for writing crypto; other things that compile directly to native code and offer C-spirited APIs for allocation and the like might be better. Having no allocation API that doesn't zero returned buffers is somewhat correlated with being high-enough level that it's a more-bad language to write crypto in.

In other words, there may not be a good language, but some languages (or runtimes, really) are certainly worse than others.