"If it gets blown into a long-term thing, I think it could potentially have a real impact," said Steve Massocca of Wedbush Securities. "In the meantime, I view it as a lot of posturing for the next election."
Funding the government was the first hurdle ahead of what is anticipated to be a bigger tussle over the $16.7 trillion debt limit, which will be reached Oct. 17. The Obama Administration has said the ceiling must be raised to fund expenditures that have already been approved by Congress, but many Republicans disagree.
(Read More: Here comes the DC shutdown: What you need to know)
"People were saying this was a warm up to the other," said Art Cashin, director of floor operations at UBS. "But the bitterness that comes out of this could make the other more bitter."
The Dow was down 128, at 15,128 Monday, but still ending September with a solid 2.2 percent gain and a gain of 1.5 percent for the third quarter. The S&P 500 was down 10 at 1681, and it was up nearly 3 percent for the month and 4.7 percent for the quarter.
"If they do shut it down tonight, tomorrow could be ugly in the morning, and we'll wait to see where we go from there," said Cashin.
(Read More: Shutdown looms, market ignores impact)
The drama in Washington also takes place on the cusp of a month that has been notoriously volatile for markets and could be again,especially after the relatively smooth sailing and gains of September. The CBOE's Volatility Index, or the VIX, jumped 7.4 percent Monday to 16.60. The VIX is watched as a measure of market fear.
"It should be an interesting ride," said Dan Suzuki, U.S.equity strategist at Bank of America Merrill Lynch. Suzuki said it's very likely Congress could spark a market sell off that would create a buying opportunity.
"If you see a five or 10 percent pullback, if you look at valuations, investor sentiment and the fundamental outlook, you want to be on the side of taking advantage of those sell offs in the market," he said. "I think it will be a buying opportunity. Part of that is conditioned on what is a low probability event that we trigger the debt ceiling. I think there's just too much pain for that to be a likely outcome."
Many traders and analysts were bracing for a short shutdown, and before the past weekend, many did not think the government would be shut down at all. "There's still an extremely low risk to long-term damage to the economy and the markets. There's no reason to get fired up about it yet,"said Bill Stone, chief investment strategist at PNC Wealth Management.
(Read More: Market has more to worry about than shutdown)
Stone said the stock market has typically suffered only a modest downturn before, during and immediately after shutdowns. In the past 17 shutdowns, the S&P 500 declined an average 0.2 percent during the shutdown and then mostly rebounded within 10 days after the government reopened. Of the six shutdowns that lasted more than five trading days, the S&P 500 fell a median 2 percent.
The last two shutdowns were in 1995 and 1996. The S&P was flat during the last 21 day shut down but lost 2.7 percent in the 10 days after it ended in January, 1996.
(Read More: Live blog: Final hours before shutdown)
"Each day that goes by, there's a greater propensity for stocks to sell off and the dollar to get weaker," said Dennis Gartman, publisher of the Gartman Letter. If Congress were to avoid a shut down or if it a shutdown ends quickly,"stocks soar, the dollar gets stronger again. Gold may event get stronger again,"he said.
Guy LeBas, chief fixed income strategist at Janney Montgomery, said the Treasury market has priced in a short shutdown, with the 10-year note yielding 2.61 percent Monday. "A quick shutdown maybe shakes market confidence and economic confidence a little, but I don't think there's that much material impact. It's very easy for Congress, after a few days of shut down to go back and pay all past bills," he said.
A three week shutdown would mean that $4.3 billion in pay lost by government workers.
He said if a shutdown were to last longer, like the 21 days in 1995 and 1996, it could shave an estimated 0.9 percent to 1.4 percent off fourth quarter economic growth. "That would take away the bulk of fourth quarter growth. That's a club in the back of the head of the economy,"he said.
LeBas said the Congressional bickering could send the 10-year yield down to 2.5 percent. "I think the bond market is much more concerned about what goes on with the debt ceiling debate. Its got a chance to rattle a lot more markets as we go into that debate. I think we'll continue to see a risk off trade," he said.
Marc Chandler, chief foreign exchange strategist at Brown Brothers Harriman, agrees the 10-year yield could move to the 2.5 percent level, and the dollar should also stay weaker. "Because of the debt ceiling, the issue is preventing us from looking at Europe more closely," he said, noting the euro is likely to stay stronger against the dollar despite political issues in Italy and fiscal problems in Greece.
"European problems will come to the fore later. That means we could see a weaker dollar. Our issue is going to be fully resolved by the end of October, one way or the other. Europe's problems are more protracted nature.It's not clear a new German government will be in place at the end of October," he said.
Germany is still working to forge a coalition government after the elections this month. "It's holding up aid for Greece, and a single supervisory system" for banks, Chandler said.
Analysts see little chance the government will default on its debt if the debt ceiling is breached. LeBas said there are mechanisms in place to avoid a default, including the U.S. Constitution. The other concern, however, is that the U.S. credit rating would be downgraded again, after the initial cut of its AAA rating by Standard and Poor's in 2011.
"There's a head-scratching irony that can only happen in finance, that the Treasury yields are falling on this debate," he said.
—By CNBC's Patti Domm. Follow here on Twitter