Michael MacAskill's answer has talked of limitations of Dragon and the SpaceX space suits, but there is a separate issue with achieving intercept.
Near Earth Objects have broadly similar orbits to earth so achieving a transfer to one from earth escape velocity is low delta V. The key part is that you must achieve escape velocity first, which is less than it took from earth surface to ISS orbit but not by much (3kms LEO to escape vs 9kms surface to LEO). Dragon and similar capsules have hundreds of meters per second DV left once in LEO, not kilometers.
If we go 'aha, I will wait till one of these objects is near earth' we run into the issue that this object will have been accelerated as it comes in towards earth and be moving faster, so we still need system escape velocities on our rocket and now have the problem that our visit time is in minutes or possibly hours if we do not want to spend years on the thing.
The last point is because NEO are actually tricky to do return trips to because their orbits are so similar to earth. If we do a manned mission to an NEO in a 1.2 year orbit at closest approach, it will be five years until next close approach, and early return will involve a very non optimal trajectory. In some ways Mars is easier since return windows happen more often.
What Dragon and similar capsules certainly are capable of is intercepting a NEO as it happens to pass near Earth for a flyby. The issue, and reason it has not been attempted is that the relative velocities will be so high (capsule slow at peak of a lofted orbit, asteroid having accelerated during approach to Earth) that the relative velocities will be at least 3 kilometers a second so humans would not see anything beyond maybe a blur, and digital sensors not doing much better.
What may be more relevant is seeking 'mini moons' objects which will have an interaction with the moon that does a reverse gravity assist and places them on chaotic orbits in the Earth/Moon system. These would be reachable by any vehicle capable of doing a moon mission and allow potentially a couple of weeks or months on site before having to return due impending ejection or collision.
Issue here is that this type of object is rare, random and small (meters), meaning the a mission would have to be prepped and kept on standby, potentially for years until one large enough to justify the trip was detected early enough to allow mission to execute.