A government shutdown doesn’t save any money. Even the bureaucratic employees that get furloughed still get their back pay once the government is running again. Shutting down the government is only a political stunt used by both parties.
It happens because the government is not ran on a sound budget of any kind. So Congress must pass spending legislation every few month that pays for the operational costs of running the country. And if there is a conflict between the political parties in Congress or with the Administration. A shutdown happens. Usually because something is or is not included in the current spending bill.
These standoffs can last days or even weeks, but generally not longer than that. Essential government services aren’t closed while others give furloughs to employees and operate on limited staffing and hours. Any complete closures are done as a political stunt to influence public opinion on the shutdown. Generally casting blame on one party or the other.
For those that are on the government payroll there can be a delay in receiving funds. This includes those on government assistant programs. But all funds are backdated so they will be payed in full once the government is back in full operation. And for those who might be facing a hardship due to the delay can find ready assistance since they will eventually be paid.