A gold standard does not mean you have that much gold in a vault, it simply means the dollar would be tied to a specific value in gold and backed by an equal amount of wealth. You need only enough gold (or silver, copper, or other real things of value) to trade for the dollars of those who wish to cash them in. Similarly a bank does not keep in a vault all the dollars that are on deposit, but merely enough to provide dollars for those who wish to retrieve their deposits. The key is to tie money, which is inherently a place holder of no inherent value, to something of real inherent value in such as way as to prevent bankers or politicians from inflating currency for their own benefit. Although if our government were not so incompetently operated (by both parties) so that we have massive debt and far too much government spending, this would not be an issue, as our currency would tend to appreciate or at least remain stable.
The reason all nations have left the gold standard is not a flaw inherent in the gold system, but a flaw inherent in people - the desire to inflate our way out of short-term problems by trading them with long-term problems. Eliminating the Federal Reserve will only trade one problem (currency controlled for the benefit of bankers) for another (currency controlled for the benefit of politicians.)