Well, if you are traveling for business or in some other arrangement where someone else is paying for airfare, achieving elite frequent flyer status in a program is quite cheap. Being the travel companion of someone who has access is another route.
Somewhat more seriously, flying intercontinentally in a premium cabin (or in a few selected services, such as United p.s.) will also generally get you a same-day lounge invitation. Of course, it's only a bargain if you're on a frequent flyer award ticket or upgrade and comparing it to the full price of the ticket. Also, you may have to live with a lesser lounge if you are on an award as opposed to a revenue fare on some airlines; I want to say Singapore Airlines is one of the persnickety ones.
Some credit cards come with lounge access. The American Express Centurion card gets you into most airline lounges; holding an AmEx Platinum card gives you Priority Pass Select access. Chase has a card that will give you United Club access. That said, the annual fees on these cards is quite substantial, comparable to the cost of purchasing a separate lounge membership. They are a value only if you already have a card, if you are switching from a comparable card that lacks lounge access, or if someone else is paying the annual fee (e.g. your company for a business card).
Depending on your travel patterns, it may be worthwhile to get membership in a frequent flyer program with more liberal rules about status. For example, Star Alliance Gold generally gets you access to any Star Alliance business class lounge when traveling on any Star Alliance flight.* On United Airlines it takes 50,000 elite qualifying miles in a calendar year to achieve Premier Gold, which is their lowest Star Alliance Gold level. On the other hand, to achieve Gold status on Aegean Airlines (also Star Alliance Gold level), you need only 20,000 EQMs. Overall, if you mostly fly United, accruing to the Aegean program is a bad deal, as you will not be able to take advantage of United promotions or United program benefits like Economy Plus access or upgrades. But if lounge access is the main thing you care about, and you book fares that accrue at least 40% EQMs for flight mileage, this is a faster way to get it.
* Note that Star Alliance Gold lounge access is restricted if your status is in a US-based programs, i.e. United (and formerly Continental) and US Airways. You cannot use a United Club as a United Gold Premier unless you are traveling on a same-day international itinerary. US Airways is even more restrictive, as they disallow lounge access to Gold Preferred members for most Caribbean and Central American flights as well. Similar restrictions apply for American and Delta, in the other two major alliances.
Another option that may work is to purchase one-time airline lounge passes— from someplace other than the airline. United charges $50 for a one-time entry into a United Club, which pretty much anyone who has been in almost any United Club will tell you is a colossal waste of money. However, sometimes people get passes as gifts or other compensation who don't want or need them, and they may give them away or try to sell them. Depending on the type of pass and the airline in question, the pass may not be transferable, but for the ones that are, you'll be able to shave a few dollars off entry by getting one from Craigslist or eBay.
Yet another area to investigate are low-cost lounge offerings. I haven't flown them in years, but Bangkok Airways allowed anyone holding a ticket access to their lounge. Of course, the price of that was built into the ticket, and you expected to pay a little more than say AirAsia. Some airports offer non-affiliated lounges, such as the Airspace Lounge at BWI which starts at $17.50.
None of the above options is particularly cheap, but if everyone off the street could get into the lounge cheap and easy, there would cease to be any point going there— it would be as noisy and crowded as the terminal.