On the Federal Reserve
First off a technicality; there isn't a Federal Reserve bank, but 12 Federal Reserve banks that together make up the Federal Reserve System.
The Federal Reserve System acts as the central bank for the United States Dollar and as such takes control of that currency. The Federal Reserve Banks have a number of responsibilities, but it is indeed possible to manage without them. They were established in 1913 and prior to that the US managed without, albeit with some problems.
An important, but mundane activity that a central bank carries out it to allow bank transfers to occur. The "Reserve" part of the name refers to the reserve accounts that U.S. banks are required to hold with one or more Federal Reserve Bank. These may be thought of as current accounts for banks. It is only by having such accounts with a reserve bank, that money may be transferred from one bank to another. When bank A wishes to pay bank B, they simply instruct their local Federal Reserve Bank to move money from one account to the other. Prior to Reserve Banking one would have to physically ship the money from one bank to the other. Whilst excellent for generating movie plot lines, this behaviour is not ideal for the banks.
Another job of the Federal Reserve System is to act as lender of last resort. This ensures that there is always liquidity available in USD, so that trades can be settled at the end of the day. This can be an important tool for preventing contagion of bank runs, as well as allowing banks to survive short term exogenous problems, such as the September 11 attacks. On the flip side, there is the risk of Moral Hazard; that banks are more reckless because there is support available. Historically there were many more bank failures in the era before the Federal Reserve, and little evidence to suggest that they were any less reckless. In the period 1837-1862 the average life span of a bank was 5 years.
Immediately prior to the Federal Reserve System, much of these two roles were taken up by private banks. A system of "National Banks" were in place, which held accounts for smaller regional banks, in much the same way the Federal Reserve Banks would later do. Similarly institutions such as the New York Safety Fund would provide deposit insurance, while others acted as pseudo lenders of last resort.
One irritating consequence of having no central bank, is that the value of currency depends upon the bank that prints it, and the bank in which your account is in. So prior to the Federal Reserve, buying a cart might cost \$10 Acme Bank Dollars, but \$11 Zenith Bank Dollars. With the various bank notes circulating together, this does not help efficient trade.
The Federal Reserve System was established to deal with these issue, which by and large it has done.
The other major activity of the Federal Reserve System, are the open market operation. That is, it controls the interest rates in an effort to "look after" the economy. This is a very 20th Century thing to be doing, and really took off after the end of the Breton-Woods system. It is difficult to objectively answer how successful this has been, but it is only one small part of the job of the Federal Reserve Banks.
Can you pay off the debt?
This question is roughly unrelated to the existence of the Federal Reserve System or not. My guess is you refer to this apparent paradox: Currently the total US debt is around 17 trillion. The total money supply (M2) of the US is around 11.6 trillion. At first glance this looks odd - the U.S. has borrowed more money than exists.
In fact, this is not a problem at all, and is just a quirk of numbers with no real significance. For starters, of that 17 trillion, 5 trillion is "intra-government debt"; that is where one government department owes another government department. So that could instantly be paid off in theory. This is just like if you put some money aside each month to buy a new car, but then "borrow" some money from your car fund to buy a TV.
Really the US debt is around 12 trillion, a tad over the money supply. The money supply is fairly arbitrary though, and is a measure of LIQUID money. The value I am using is roughly the sum of:
Cash, Coins, Demand Deposits, Checks, Savings Accounts, Money Market Accounts/Funds & certificates of deposit under \$100,000
Obviously, there is a lot of wealth NOT included in that. The total value of the US Stock market is around \$16 trillion, the total value of corporate bonds is around \$10 trillion, and the mortgage related market comes to something around \$10 trillion. Then there are the buildings, resources, land etc. Adding everything up at current market values and the entire united states is estimated at around \$123 trillion: there is no shortage of wealth.
So really the questions is how do you pay back a debt without using up the money? And the simple answer is that money is not destroyed when the debt is repaid, it simply moves from one person to another. So if the US pays off \$5 trillion of debt, then there will STILL be \$12 trillion of money supply, it will just have changed hands.
Alice owns \$1000 of US Debt
Bob owns \$500 of US Debt
Chris runs a business and has money in his account (part of the money supply). He pays \$1000 of taxes to the government.
The government pays Alice \$1000 to pay of the debt, which goes into her account (part of the money supply)
Alice spends this \$1000 at Dave's shop and it goes into his account.
Dave pays \$500 of taxes from his current account.
The government pays Bob \$500 which goes into his current account.
In this example, \$1000 of money pays of \$1500 of debt, and ends up with \$500 sat in Dave's account and \$500 sat in Bob's account, ready to carry on the cycle.
The only impediment to paying off the debt is raising sufficient money to have a budget surplus with which to pay back the debt.