The backup menu is in the Administration Panel. And in the Left menu, inside the Server Administration, go on Backup.

TeamCity (as of v10) does not automatically backup, but you can get TeamCity to back itself up on a daily basis by scheduling a task to hit the REST api. Typically you would also need to schedule a second task (an hour? later) to copy the finished backup off the main TeamCity server and to a safer place.

(Note for security reasons it is better not to run an agent on the same machine as the TeamCity server)

First of all, Ensure that the user which will be running this call has the Change backup settings and control backup process privilege.

# 
# TC Backup Launcher
# Script to launch a backup on the TeamCity Server
#
Param(
    [Parameter(Mandatory=$true)][string]$username,
    [Parameter(Mandatory=$true)][string]$password
)
Begin
{
    $url = "http://teamcity:8111/httpAuth/app/rest/server/backup?includeConfigs=true&includeDatabase=true&includeBuildLogs=false&fileName=TeamCity_Backup"
    $username = <username>
    $password = <password>
    
    # Function to realize a POST Operation
    function Execute-HTTPPostCommand() {
        param(
            [string] $target = $null
        )
        $request = [System.Net.WebRequest]::Create($target)
        Write-Host "POST: " $request.RequestUri
        
        $request.PreAuthenticate = $true
        $request.Method = "POST"
        $request.ContentType = "application/xml"
        $request.Headers.Add("AUTHORIZATION", "Basic");
        $request.Accept = "*"
        $request.Credentials = New-Object System.Net.NetworkCredential($username, $password)
         
        $response = $request.GetResponse()
        $xmlout = ""
        
          if($response)
          {
              $sr = [Io.StreamReader]($response.GetResponseStream())
              $xmlout = $sr.ReadToEnd()
          }
        return $xmlout;
    }
    
    Write-Host "Creating a new Backup:"
    
    Execute-HTTPPostCommand $url
}

The important part is the URL to call with method POST :

http://teamcity:8111/httpAuth/app/rest/server/backup?includeConfigs=true&includeDatabase=true&includeBuildLogs=false&fileName=TeamCity_Backup

You can specify the settings you want to backup as in the web interface.