To be "on your last leg" is to be near the end; near collapse. You have scarce resources left to complete something. "Our old family dog is on her last leg (meaning tired, near death). The "leg" here can be interpreted in two ways. First, comparing a leg to the tired or injured legs of a person or an animal after extended effort; there is not much left to stand on.  A "leg" is also a counting word for a segment of a long journey. "I would be glad to sell you my car, but I must tell you that it is on its last leg."