Matrices can be everything between dense and extremely sparse, depending on the application (sometimes you have a structured matrix and can use sparse methods to reduce it to a dense system of much smaller size). But yes, dense matrices with thousands of rows do occur. The applications where Strassen's asymptotically fast multiplication algorithm really helps are certainly rare compared to overall uses of linear algebra, but they do "strictly" exist.

Strassen multiplication usually starts to win over classical multiplication already for matrices with a few hundred rows (in special circumstances, it is even superior for 2x2 matrices). The Coppersmith–Winograd algorithm, on the other hand, is probably "strictly of theoretical interest" for the time being...