Can one weight train without gaining too much muscle?

Yes: Most women will not get too much muscle by weight training because women have very little testosterone, the hormone that helps build big muscles. If a woman uses lighter weights and does more repetitions, instead of doing fewer repetitions with heavy weights, she will get strong without getting huge muscles (the same concept works for men also).

Low weight high reps: Classically, lower weight and higher repetitions will yield improved strength and endurance with less emphasis on actual muscle mass gains.