This is a far-reaching question, and an important question is your role. Do you work for the IT department in a medium/small company or something else?
Any group that runs one or more service should have an SLA (Service Level Agreement) with the business owners, which spells out when the system should be available, what downtime that is permitted and when, how long time it make take before the system is
back up after an unexpected failure etc.
In your situation it seems that you had quite an easy ride, as weekends were available for system maintenance. Now the business wants to use the system on weekends. This means that you need to review your SLA, and most likely it means that your departments
needs a bigger budget.
Olaf mentioned online reindexing. This feature is only available in Enterprise Edition, and I would guess you have Standard Edition. So there is a cost right there. I assume that your maintenance work also includes time to apply Windows updates and similar.
To avoid downtime you would need something like a cluster, so you can apply fixes on the cluster. This is likely to mean that you need to buy more hardware. Administrating a cluster is more complex than a plain server, so there is a cost for training.
What if, the business is not prepared to cough up the money for this? Simple. Then you stick to your current SLA and users will have to accept that access on the weekend is not guaranteed.