I recently purchased a home about 25 years old. I didn't realize it at the time but my crawl space has an uneven dirt flooring. There is evidence of busted up concrete around supports.(I am not sure if the crawl flooring was originally constructed of cement). A very thick gauge of plastic has been placed over the dirt flooring( the dirt loooks like clay, it's very hard). I would like some pro and con suggestions concerning covering the dirt floor with something more sturdy and to minimize the possiblity of insects. I would to like use the crawl for storage as there is no evident of water seepage. There are two vents located in different corners of the south wall. My living room and dining area is located above the crawl area.

 

Crawlspaces normally have dirt "flooring".
The plastic on it is a vapor barrier to reduce ground moisture under the house.
The vents are to provide some airflow to help prevent mold, mildew, wood rot, etc. from excessive ground and air moisture.
The concrete around the support piers are the pads that the piers are set on.
The only way to control insects (spiders, etc.) under the home is to spray it periodically with an off-the-shelf bug spray.
Only a professional, licensed pest control operator can properly treat for termites, powder post beetles, etc.
Everything sounds normal to me.
You can store anything under your house, but be aware that if it can be damaged by air/ground moisture, mice, etc., it WILL be.
You can lay pressure-treated boards or 4X8' sheets of p-t plywood on the ground for storing things on. I would NOT put anything under there touching the ground that is not termite resistant.
Good Luck!