A load balancer that fails takes down everything behind it. You need redundant load balancers.
Active-passive: One balancer handles traffic. The standby monitors. If active fails, standby takes over using a floating IP address.
Active-active: Both balancers handle traffic. DNS returns both IPs. If one fails, DNS health checks remove it. Clients use the survivor.
Cloud providers handle this automatically. AWS ELB runs across multiple availability zones. You get redundancy without managing it yourself.