Many people said they couldn't retrieve emails from their inboxes or complained that emails were not being delivered.
"We are continuing to work on resolving this and ask that you stay tuned for further updates," Yahoo said in a message fired off on Twitter.
The California-based firm on Wednesday apologized and promised a fix by the end of that day, but problems have persisted.
(Also see: Yahoo restores email access to most users, starts updating inboxes with backlog)
Yahoo would not disclose the number of email accounts knocked out of service since late Monday night California time. It is estimated there are about 280 million Yahoo Mail users.
"We are very sorry for recent difficulties with Yahoo Mail," the company's senior vice president of communication products Jeffrey Bonforte said in a short message posted on Tumblr.
He blamed the outage on a "hardware problem" at one of Yahoo's data centers.
Some functionality was returning to the free Web-based email service as Yahoo scrambled to get it back in full operation.
Yahoo Mail users turned to Twitter and online chat forums to unleash frustration at being cut off from email and threatening to switch to a free service offered by Google.
Among complaints was a woman who complained having emailed invitations to a holiday party and not knowing what to expect because replies aren't being delivered.
"I need my account back!" a woman identified as Natalie Bean said in a message on Twitter.
"Fix it so I can permanently switch emails!"
The Yahoo Mail disruption marks a stumble for chief executive Marissa Mayer, who has made focusing on people's "daily habits" such as email part of her plan to revive the faded Internet pioneer.
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