Yes. Systems programming and Embedded are your best and most visible playing fields, but many large, legacy applications were written in C and continue to be maintained.

> Is it possible for C to be a standalone skill,

No. As others have pointed out, the language + standard library is very spartan and will only take you so far. This will only get you an entry level position, and only in teams that are big enough to have some senior people with spare capacity for mentoring, and a stream of small, self contained tasks for you do while in training.

To be able to work independently you need to have at least some basic knowledge of the whole toolchain: Compiling (you need to know to heart the different steps that are taken by the compiler, and at least its 20% most common cmdline flags), Building (make), program analysis (lint, valgrind), debugging (gdb, or whatever comes with the compiler you are using), 3rd-Party-Libraries (pick 2-3 of: glib, pthreads, antlr, curses, openssl, etc), Standards (MISRA, POSIX - which is at least as much about the API to Unix-like OS as it's about the C language).

From there, there are more tools to help you, but those are typically OS dependent and are not exclusive to C.

C doesn't exist in a vacuum: it has to run on something. And the standard library doesn't get you very far.

You'd need to know at least one OS API as well: POSIX is probably best, maybe Win32, an embedded OS/executive might work also, or perhaps even a bare-metal CPU or two.

C + POSIX covers a lot of the Open Source world, and increasingly more of the embedded market. C + VxWorks is possibly the next best combo.