It is true that, in their purest forms, the two approaches are mutually exclusive, since socialism is essentially based on public ownership, while capitalism, in its most extreme form, is based exclusively on private ownership.
However, it is entirely possible to implement elements of socialism within a capitalist market economy, whereby key parts of socially important institutions remain under state control while others are organized privately.
One example of such an approach is the social market economy, which plays a role in Germany, Austria, and Switzerland.
It was developed among others by the economist Alfred Müller-Armack and seeks to “combine the principle of freedom in the market with that of social balance.” This approach, in turn, bears a certain resemblance to ordoliberalism (as advocated by Walter Eucken, among others), which had a formative influence on German-speaking countries in Europe after World War II.
Unfortunately, however, this approach—which I consider sensible—is not immune to U.S.-style neoliberal capitalism (Chicago school of economics and others), which has, regrettably, also taken a firm hold in Germany today and whose effects are, among other things, significantly evident in the fact that the wealthiest segments of the population are becoming ever richer, while social benefits that benefit the general public are being cut back further and further.
In my view, however, this is less a problem with the approach itself and more—as with all theories—due to the fact that any system can be corrupted by unscrupulous people.