I have a cheap metal filing cabinet - the flimsy kind you get at office depot. I hang my tissue to dry and when they feel right I put them into plastic ziploc bags and then into the filing cabinet. I take them out as I use them in the printing session. The ziploc bags prevent them from drying out more as time goes by. This is not long term storage - only for the printing session.
Edit - I use alcohol or acetone to speed the drying of the tissue. Mine will dry in 2-3 hours. If you use thick tissue you may need much longer even with alcohol or acetone