Can I have my own self-hosted build agent server in Azure Devops, with fallback to cloud?

Is it possible to create an Agent Pool with self-hosted agents - which falls back to Azure Pipelines in case the self-hosted agents are unavailable?

I tried searching for an answer for it and poked around in Azure DevOps. It does not seem to be possible.

That requires the pipeline to know if there is any idle agent in a self-hosted agent pool before assigning the agent. For this, you may have a test to call the REST API request in the agentless job pipeline task of InvokeRESTAPI@1 to check if the agents in an agent pool are assigned with requests, then determine to run the subsequent job in a MS-hosted or a self-hosted agent pool.

parameters:
- name: agentIds
  type: object
  default:
  - 465
  - 547

variables:
  system.debug: true
  orgName: ${{split(variables['System.CollectionUri'], 'https://dev.azure.com/')[1] }}
  poolId: 1 # Default
stages:
- stage: StageCheck
  jobs:
  - job: AgentlessJob
    pool: server
    steps:
    - ${{each agentId in parameters.agentIds}}:
      - task: InvokeRESTAPI@1
        inputs:
          connectionType: 'connectedServiceName'
          serviceConnection: 'AzureDevOpsServices'
          method: 'GET'
          headers: |
            {
            "Content-Type":"application/json", 
            "Authorization": "Bearer $(system.AccessToken)"
            }
          urlSuffix: '/$(orgName)/_apis/distributedtask/pools/$(poolId)/agents/${{agentId}}?includeAssignedRequest=true&api-version=7.1-preview.1'
          waitForCompletion: 'false'
          successCriteria: 'eq(root.assignedRequest.reservedAgent.id, ${{agentId}})'
        continueOnError: true

- stage: StageOnSelfhostedAgentPool
  dependsOn: StageCheck
  condition: eq(dependencies.StageCheck.result, 'SucceededWithIssues')
  variables:
    stageDependencies: $[convertToJson(stageDependencies)]
  jobs:
  - job: AgentJob
    pool: Default
    steps:
    - script: |
        echo "Detected idle agents in the Default self-hosted agent pool"
        echo "$(stageDependencies)"
      displayName: Run agent job in Default self-hosted agent pool

- stage: StageOnMShostedAgentPool
  dependsOn: StageCheck
  condition: eq(dependencies.StageCheck.result, 'Succeeded')
  variables:
    stageDependencies: $[convertToJson(stageDependencies)]
  jobs:
  - job: AgentJob
    pool:
      vmImage: ubuntu-latest
    steps:
    - script: |
        echo "No idle agents in the Default self-hosted agent pool"
        echo "$(stageDependencies)"
      displayName: Run agent job in MS-hosted agent pool

Besides, you can probably go through the features of Managed DevOps Pools and VMSS Agent Pools, which may be suitable for you.