A Hami melon is actually a cultivar of cantaloupe, although it doesn't look like the cantaloupes we all know and love.  Since it's a cantaloupe, you can tell if it's ripe for harvest by smell (slightly sweet), and it has firm flesh that gives a little under pressure.  However, the most reliable way is to look at the stem--as the fruit gets nearer to being ripe, the area of the stem near where it's attached to the fruit develops small cracks.  When the tiny cracks encircle the stem, and the stem itself looks shriveled, the melon should break from the vine with a slight twist.  If it doesn't come off easily, give it a couple more days.

These melons can be stored for a several weeks to a few months with little loss of quality.  You may be able to enjoy your Hami melons into the winter!

Here's a good article from Oregon State about melons, and Hami melons are covered in it:  http://horticulture.oregonstate.edu/content/melons-cantaloupe-muskmelon-honeydew-crenshaw-casaba-etc...