I know they played in The Cavern Club and in different clubs in Hamburg, but I don't know how popular those venues were or what was their capacity. What's more, I don't even know what a big audience would be for them at that point.
Broadly speaking, a 'beat group' like the Beatles in 1961 (before the Merseybeat sound became a national phenomenon, and well before Beatlemania and the British Invasion etc), were almost always a local phenomenon. One thing to remember about the Beatles and indeed music at this point in general is that in 1962 there was no DJ culture. Typically, if you wanted to go somewhere at night to dance or to simply hang out with musical accompaniment, there was probably a band playing that music rather than simply recorded music over the PA. And the Beatles, in their pre-Parlophone days with Pete Best on drums, simply played what was then simply the most popular dance music among young people - rock & roll.
The Beatles did not have an enormous local reputation in Liverpool before they had toured to Hamburg for the first time - according to Mark Lewisohn's Tune In book, they weren't particularly strong rock'n'roll instrumentalists before then, and were notable chiefly for their harmony singing. As a group, they'd never had a consistent drummer until Pete Best, and he was basically someone they hired for Hamburg, rather than a consistent member of the band before then. They initially played in a fairly small nightclub in Hamburg, the Indra Club. As they gained experience playing, and grew a reputation in Hamburg (and as the Indra closed due to noise complaints), they were moved into the bigger Kaiserkeller, a club that still exists today. Over the course of their trips to Hamburg, the Beatles went to bigger and bigger clubs - the Top Ten and then the Star Club, which was also a venue where international touring acts would play (the Beatles would meet Little Richard when he toured there, and Jerry Lee Lewis recorded a well-regarded live album at the Star Club in 1964). The Beatles, at the Star Club, weren't quite star attractions quite in the same way as Little Richard or Jerry Lee Lewis (apart from when they played there after being signed to EMI), but they were effectively, during their engagements there, something like the equivalent of a venue's house DJ.
Within the UK, the Beatles had no real reputation outside of Liverpool before they were signed to EMI/Parlophone. The British entertainment industry, broadly speaking, was strongly London-centric, and the Beatles coming from Liverpool were reflexively seen as a local phenomenon that probably wouldn't translate. This 'beat group' phenomenon, in England, was commented on in the media, with an article in Disc in 1961 wondering somewhat presciently why the beat groups' local success hadn't translated to record deals and hit records.
Within the Liverpool scene, the Beatles, after their return from Hamburg, became relatively quickly established as one of the most popular bands in Liverpool. For example, the very first issue of the fortnightly Mersey Beat magazine in July 1961 sold 5,000 copies in Liverpool, and a biography of John Lennon was on page 2. Especially after they had contracted Brian Epstein as their manager in 1961, the Beatles often began earning more money from a single gig than their parents with their day jobs would earn in a week - which is obviously a measure of their consistent popularity. The Cavern itself was not an enormous venue - you can see 1962 footage of the Beatles playing there here - but it had the advantage of being the venue in Liverpool's centre that did the best lunchtime trade, where young office workers could go and see some music over lunch, and so their regular shows at the Cavern were both a convenient extra money source, and a favourite venue of the band (the Beatles would, effectively, conduct their regular rehearsals at the Cavern before the crowds came in) - but the Beatles would play at multiple other venues in Liverpool that were larger - and that had to be, given the fees for the Beatles' services that Epstein was charging (and getting).
But outside of Liverpool, the Beatles had very little reputation. For example, in December 1961, the Beatles played a show just outside of London, in Aldershot. Despite an enthusiastic promoter, they had 18 paying customers that night, who apparently mostly looked bored (this article about the show, for isntance, has some photos demonstrating the poor attendance).