The ability to create abstractions is pretty much a part of any programming language, including assembler. They're a property of the structure of the code, not of specific language features. For example, in the Linux kernel, which is written in C, there's an abstraction called a file descriptor that represents a stream of bytes that your program can read/write, whether it comes from a file on a disk, a serial port, a pipe, etc.
Interfaces also exist in all languages: they're just a well-defined set of calls that one piece of code uses to communicate with another piece of code. For example, the C runtime library (or any library) has an interface (API), and callers of the library don't need to know about the internal structure of the library to be able to call its services. (There's a pretty good book called "C Interfaces and Implementations" that talks about how to write reusable code in C.)
Classes are also possible to implement in C. A class is pretty much a data structure with a bunch of functions (the methods) that manipulate its data. In C, this can be represented by a structure and bunch of function pointers. The "constructor" function calls malloc() to create an instance of the class, sets its member variables from its parameters and returns a pointer to it. (Implementing inheritance is a bit tricky.)
To be able effectively split development tasks across a team, the same rules apply as in any other language: implement functionality behind well-defined APIs so that one part of the code can be used by another without having to know the details of how it works.