You can run a backup of all SSRS reports using the following script.

Assume you are using MSSQL 2019, locate the RS.exe Utility in below folder.
  

     C:\Program Files\Microsoft SQL Server Reporting Services\Shared Tools


For older version, try look under below folder.

     C:\Program Files\Microsoft SQL Server\150\Tools\Binn


Using Notepad, paste and save the following procedure as a "SSRS_Backup_Reports.rss". Sample file attached.

===============================================================================

Public Sub Main()
'--------------------------------------------------------------------------------------------------------------------
' Purpose: Script to backup reports from a folder on ReportServer
' Save file as .rss extension and run using rs.exe from command line.
' Reference: http://bhushan.extreme-advice.com/back-up-of-ssrs-reports-using-rs-utility/
' https://docs.microsoft.com/en-us/sql/reporting-services/tools/rs-exe-utility-ssrs?view=sql-server-2017
' Example: rs -s http://localhost/reportserver -i D:\Scripts\Backup_Reports.rss -e Mgmt2010 -v backupFolder="D:\Scripts\BackupReports" -v parentFolder="/IndividualReportFolderNameHere"
' rs -s http://localhost/reportserver -i D:\Scripts\Backup_Reports.rss -e Mgmt2010 -v backupFolder="D:\Scripts\BackupReports" -v parentFolder=""
'--------------------------------------------------------------------------------------------------------------------
Try
rs.Credentials = System.Net.CredentialCache.DefaultCredentials
Dim items As CatalogItem() = Nothing

If String.IsNullOrEmpty(parentFolder) Then
items = rs.ListChildren("/", True)
Else
items = rs.ListChildren(parentFolder, False)
End If

Console.WriteLine()
Console.WriteLine("...Backup Started...")

For Each item As CatalogItem In items
If item.TypeName = "Report" Then
Console.WriteLine(item.Path)
Dim reportPath As String = item.Path
parentFolder = Path.GetDirectoryName(item.Path) ' comment out this line to save the reports in one folder
Dim reportDefinition As Byte() = rs.GetItemDefinition(item.Path)
Dim rdlReport As New System.Xml.XmlDocument
Dim Stream As New MemoryStream(reportDefinition)
Dim backupPath As String = Path.Combine(backupFolder, Date.Now().ToString("yyyy.MM.dd") + "\" + parentFolder)

If (Not System.IO.Directory.Exists(backupPath)) Then
System.IO.Directory.CreateDirectory(backupPath)
End If

rdlReport.Load(Stream)
rdlReport.Save(Path.Combine(backupPath, item.Name + ".rdl"))

Console.WriteLine(item.Name + ".rdl")
End If
Next

Console.WriteLine("...Backup Completed...")
Console.WriteLine()

Catch e As Exception
Console.WriteLine(e.Message)

End Try

End Sub


===============================================================================


Create below subfolders in C:


C:\Temp

C:\Temp\SSRS

C:\Temp\BackupReports


Save the "SSRS_Backup_Reports.rss" in folder C:\Temp\SSRS


Open DOS Command prompt with Run-as-Administrator. 


Change folder to RS utility folder with below command:


     cd C:\Program Files\Microsoft SQL Server Reporting Services\Shared Tools


Paste below into the DOS Command line. 


rs -s http://[Server Name]/ReportServer -i C:\Temp\SSRS\SSRS_Backup_Reports.rss -e Mgmt2010 -v backupFolder="C:\Temp\SSRS\BackupReports" -v parentFolder=""


Note : 

1) Please specify the correct parameter values in blue fonts.

2) Setting Parameter parentFolder="" means to save the entire folder structure with all the reports.


You can find a copy of the SSRS *.RDL file in below folder.


C:\Temp\BackupReports


More information regarding the RS Utility is available below.
SSRS RS.EXE Utility