Is there any lockers at the Hongqiao train station that I can securely store some of my luggage while I go to Beijing for a few days? Thanks!

There are lockers in the train station to store the luggages, but the luggages cannot be stored overnight. So it is impossible to store for a few days.
If you want to store for several days, you may try to store it in Hongqiao Airport and the airport is just about 15 minutes walking distance from the railway station.