Rotherham NHS Trust reportedly wrote to at least 80 workers - mostly medical staff - informing them they are required to leave the three blocks, Swale Court, Derwent Court and Loxley Court, by the end of January over fire safety concerns.
The trust said a recent assessment of the blocks had found ‘extensive work' would be needed blocks to update the 1970s-era buildings to modern fire safety standards and the costs were ‘not affordable'.
Protestors said the decision could impact on staffing levels at the hospital.
Sheffield Save Our NHS said: ‘Evicting these tenants, mostly medical staff, of their accommodations translates to a direct impact on patient treatment.
‘Practitioners who require this accommodation are vital to the functional wellbeing of our healthcare system, from rotating junior doctors to emergency staff accommodation, to attracting new colleagues.'
The protestors have also set up a petition in a bid to fight the plans.
In response, a spokesperson for The Rotherham NHS Foundation Trust, said: ‘We understand the concerns from the affected residents and have held two listening events with residents to understand these concerns. One to one meetings have been offered to all residents so their needs can be understood and solutions developed to resolve the situation.
‘The trust is committed to working with those residents affected by these changes to assist them in finding alternative accommodation and to retain them at the Trust for the benefit of patients.'
No tenants have been asked to leave before the end of their current lease, the trust added.