Monte Carlo simulations are reported for the conformational properties, static structure, and equation of state of two-dimensional polymers. The polymer molecules are modeled as chains of freely jointed tangent hard disks. For isolated chains, the scaling of the mean-square radius of gyration, R-g(2) with degree of polymerization, n, is consistent with R-g(2) similar to n(3/2). Over the range of concentrations and chain lengths studied, the chains do not behave as ideal chains, with R-g(2) similar to n(1.07+/-0.03) at the highest concentration studied. Surprisingly, for very high concentrations, the chain size increases with increasing concentration. Contrary to expectations, the chains do not become disklike at high concentrations, and the asphericity changes by only about 10% over the entire range of concentrations. In fact, the distribution of shapes of the chains remains essentially unchanged as the concentration is increased, and there is considerable interpenetration between polymer molecules. Results are also presented for the pair correlation functions and the equation of state. The osmotic pressure, p, scales with concentration, c, as p similar to c(4.7+/-10.1), which is stronger than the scaling prediction of p similar to c(3).