Is it possible to obtain single strands of DNA in solution?
There is usually a denaturation step in a typical PCR protocol, see e.g. here, which serves exactly this purpose: the strands are thermally divided (typically at 95 °C for a few seconds). As you can see, this is a very common process, but the strands only remain denatured if kept at high temperatures.

Room-temperature strand separation in regular buffer solutions will be very hard, because annealing (the opposite process, where two complementary strands pair up) is thermodynamically favourable.
More permanent separation might be obtained by dissolving DNA in particular solutions, as described here, where DMSO seems particularly effective.
This results in separated strands at room temperature, but both strands will clearly still be present. One might think about separating the two with various methods (e.g. HPLC), but I doubt these solutions are compatible with regular HPLC columns.