Get TeamCity to back itself up by creating a scheduled build configuration that runs this Powershell script.

teamcity-backup.ps1

param(
    [string] $username,
    [string] $password,
    [string] $baseUrl = "http://teamcity:8111",
    [int] $sleep = 10,
    [int] $timeout = 600
)
function Execute-TeamCityBackup {
    param(
        [string] $baseUrl,
        [string] $username,
        [string] $password,
        [string] $filenamePrefix = "TeamCity_Backup_",
        [string] $addTimestamp = $true,
        [string] $includeConfigs = $true,
        [string] $includeDatabase = $true,
        [string] $includeBuildLogs = $true, # change to false if the (very large) build log history is not required.
        [string] $includePersonalChanges = $true
    )
    $url = [System.String]::Format("{0}/httpAuth/app/rest/server/backup?addTimestamp={1}&includeConfigs={2}&includeDatabase={3}&includeBuildLogs={4}&includePersonalChanges={5}&fileName={6}",
        $baseUrl,
        $addTimestamp,
        $includeConfigs,
        $includeDatabase,
        $includeBuildLogs,
        $includePersonalChanges,
        $filenamePrefix)
    $webRequest = [System.Net.WebRequest]::Create($url)
    $webrequest.PreAuthenticate = $true
    $webrequest.Credentials = New-Object System.Net.NetworkCredential($username, $password)
    $webRequest.Method = "POST"
    return ([Io.StreamReader]($webrequest.GetResponse().GetResponseStream())).ReadToEnd()
}
function Get-TeamCityBackupStatus {
    param(
        [string] $baseUrl,
        [string] $username,
        [string] $password
    )
    $url = "{0}/httpAuth/app/rest/server/backup" -f $baseUrl
    $webrequest = [System.Net.WebRequest]::Create($url)
    $webrequest.PreAuthenticate = $true
    $webrequest.Credentials = New-Object System.Net.NetworkCredential($username, $password)
    return ([Io.StreamReader]($webrequest.GetResponse().GetResponseStream())).ReadToEnd()
}
Write-Host ("Triggering backup.")
$backupFilename = Execute-TeamCityBackup -baseUrl $baseUrl -username $username -password $password
Write-Host ("Backup filename: {0}." -f $backupFilename)
Write-Host ("##teamcity[setParameter name='backupFilename' value='{0}']" -f $backupFilename)
$status = ""
$timeoutTimespan = New-Timespan -Seconds $timeout
$stopwatch = [diagnostics.stopwatch]::StartNew()
while (($status -ne "Idle") -and ($stopwatch.elapsed -lt $timeoutTimespan)) {
    $status = Get-TeamCityBackupStatus -baseUrl $baseUrl -username $username -password $password
    Write-Host ("Backup status: {0}" -f $status)
    if ($status -ne "Idle"){
        Start-Sleep -Seconds $sleep
    }
}
if (($status -ne "Idle") -and ($stopwatch.elapsed -gt $timeout)) {
    Write-Host ("Backup execution exceeded script timeout value of: {0} seconds." -f $timeout)
    Write-Host ("Backup may still be in progress.")
}
elseif ($status -eq "Idle") {
    Write-Host ("Backup complete.")
}