Finding nonnegative solutions to an underdetermined linear system

This is a standard linear optimization problem. To summarize, the set of all nonnegative solutions can be completely described by either a finite system of linear inequalities, or a finite "generating set" such that every solution is a positive linear combination of elements in the generating set.

A $8\times 4$ matrix is very small and reasonable in this context, so if you have exact coefficients for your matrix you will be able to determine the set of nonnegative solutions exactly.

http://en.wikipedia.org/wiki/Linear_programming#Solvers_and_scripting_.28programming.29_languages

Matlab has a function lsqnonneg which can minimize $\| Ax - b\|_2$ subject to the constraint that $x \geq 0$.

The minimum value will be $0$ if and only if there exists a nonnegative solution to $Ax = b$.