Money market accounts at the nation's banks and thrift institutions soared to a total of $111 billion in the week ending Jan. 5, the Federal Reserve reported yesterday.

This makes the market rate savings account the most popular ever introduced, far surpassing the All Savers account. Since they were introduced Dec. 14 to rival money market mutual funds, money market accounts have taken in an amount equal to more than half the assets held by the funds, or $205 billion. These have lost some $27 billion since Dec. 1.

The statistics surprised most experts and alarmed some others. Richard B. Ross, vice president of Market Facts, a Chicago research firm that had predicted money market accounts would garner $70 billion by the end of 1983, yesterday revised his estimate to potentially twice that amount by the end of next month. He calculates the percentage of new money--dollars from sources outside the issuing institutions--at between 25 and 30 percent. Some economists have estimated financial institutions need 50 percent new money to break even.

Leonard Shane, a California savings and loan executive who is chairman of the U.S. League of Savings Institutions, warned that the high interest rates paid on the new accounts could delay or limit the much needed housing recovery because lenders will not want to commit funds that can be withdrawn daily to 30-year mortgages.

There also is concern that the high rates could increase an institution's cost of funds and adversely affect their earnings.

Last week economists of the American Bankers Association concluded that there was little shift in funds out of passbook accounts and into the higher yielding accounts. But an ABA spokesman said yesterday the figures would have to be revised. Commercial banks had captured 64 percent of the dollars in the new accounts at last count.

James Christian, chief economist of the U.S. League, predicted "no grave damage" to thrifts' earnings so long as the current interest rate "war" stops this month, rates decline to market levels, and the flow of funds between institutions stabilizes.

Soon after the money market account was announced in November, institutions began offering rates up to 22 percent. The average, acording to Market Facts, was 11.14 percent at time when the money market funds were paying about 9 percent. Rates have slipped generally everywhere, but there are still isolated cases of promotional rates, such as Riggs National Bank's 15 percent and Fairfax Savings' 18 percent.