That money comes from tax revenue. You already paid taxes on your income once. Now you are getting taxed on it again when it recycles back to you. You will also get taxed on it again when you spend it. There is a flow chart somewhere that shows the myriad of ways the same money is repeatedly taxed.
Technically it comes from unemployment insurance premiums. Wikipedia says: In the United States, benefits are funded by a compulsory governmental insurance system, not taxes on individual citizens.
It's easy to imagine some sort of alternate reality where the unemployment insurance system is handled by a private company, for instance.
Moreover, calling it double-tax doesn't really make any sense because the tax is on the income, not the dollar itself. Public sector employees aren't getting double-taxed because part of their salary goes to the government, which pays for their paycheck. You're also not taxed infinity-taxed because your paycheck comes from a business which is taxed, which gets its revenue from a consumer which is taxed, which gets their paycheck from a business which is taxed...
If it isn't a tax, then may I opt out of it and/or get all the money back I put into it? My current strategy is government grants to fund project I want to do regardless, but I don't think that will ever really break even.