I know that the reason I couldn't lift myself up(without the help of any items or a pulley system)was due to that my force was internal and there is a reaction force. However, assuming that somebody attempts to lift me up, wouldn't there also be a reaction force that prevents him from doing so?
You have already given the answer, it's Newtons third law actio = reactio. If you apply a force to yourself to lift you up, there is an equal force pressing you down. If another person lifts you up, then he is pressed down to the ground.
Actually you can kind of "lift up" yourself in the meaning of pressing up yourself, but only if a part of you (e.g legs or arms) is touching the ground. You are doing that all the time when walking.
Of course you can lift yourself up. Have a look at this video where a mom is using the tree house pulley chair to lift herself to the tree house.Force is a vector, it is all a matter of the direction of the force applied, not just its magnitude.