It is possible to store $CO_2$ geologically similar to natural burial. But at the rate that it needs to be buried, how do we know it is economic enough that more fossil fuels wouldn't have to be burned than what can be buried at an effective rate?
It is possible and there are several demonstration projects running. In Europe, the carbfix project quotes a price of $24.8/ton for carbon sequestration. Geothermal power is used to pump the CO2 underground, so no energy from fossil fuel is required.
There are concerns about the available capacity for storage using these mechanisms, and the pumping process does trigger seismic activity (e.g. wikipedia).
It is likely that carbon buried in geological formations will be sequestered more reliably and for longer that carbon which is simply absorbed by trees ... but there is no guarantee on that there will not be some leakage.
Because of these concerns, and because generating electricity from sustainable sources is cheaper than burning fossil fuels and then trying to manage the pollution, the main focus of dealing with global warming should be reduction of fossil fuel use to near zero.
There may well still be a significant role for carbon sequestration in dealing with some CO2 emissions which are difficult to avoid.
Of course ,but it would be very expensive and there would be the earthquakes. Consider that many blame fracking for earthquakes ; fracking lasts a matter of days then pressure is released and water/oil is pumped out. To sequester CO2 would involve pumping into the same or similar formations for years, so presumably cause earthquakes. There would be some good locations where CO2 could be pumped into an oil/gas formation as the oil/gas is removed . And there are some locations where that has been done for years ( CO2 and water pumped into the formation to "push" out the oil).There are many possibilities but any involving the large quantities of CO2 I guess you mean, would be expensive and some groups would object to.