Idea 1: For a generally approved outdoor setup, you're gonna pay some big bucks ($10K).
Idea 2: If you wanna slide the gate, you won't use a linear actuator, but a pulley and sprocket. If you wanna swing the gate, then maybe an actuator. The pulley and sprocket also requires tracks and rollers for the gate.
Idea 3: If you want remote control, you'll also want security and encryption as opposed to 492MHz that everything else is on. Otherwise anyone can get in. More $$.
Idea 4: Finally, with not-too-difficult mechanical knowledge, you could adapt a garage-door opener to slide open a gate. The combinations are endless and the cost would be in the hundreds, not thousands. The combinations are so many that such an adaptation would be difficult to converse about without being there.
Idea 5: I would almost be surprised that someone on YouTube hasn't done such a thing, already.