Yes, you can make a good looking rug with no sewing and basically no out of pocket expenses.
Arrange the carpet squares into a 7×5 rectangle.
Flip them over, making sure everything is aligned well and seamless.
Measure across the rug, then cut and peel one side of the tape. Carefully start taping the squares together in horizontal rows.
Measure and cut the shelf liner to match the size of the rug.
Peel the other side of the tapes so that the sticky side is exposed to the shelf liner. carefully place the shelf liner of top of the tape, press to adhere.
Trim excess rug squares that stick out.