When you say "without spending loads", what kind of figure are you talking? Load balancers are more useful to maintain reliability under heavy load than purely to maintain 100% uptime. There's no reason why a single decent, well-specced dedicated server couldn't achieve that target (from ~£100/month). The next step up being 2 servers and 1 load balancers still gives you a single point of failure so in terms of risk from a PSU going bang, etc you're no better off.
Edit: As for round robin DNS, again that's more useful for load balancing than for maintaining uptime. Regardless of how low you set your TTL, plenty of servers will simply ignore it meaning many of your users could continue to see the downed server for hours/days afterwards.