Sure, here's a bash script that handles web requests using only binaries commonly found in BusyBox:

```bash
#!/bin/sh

while true; do
  # Start listening on port 8080 for incoming requests
  nc -l -p 8080 <&0 | while read line; do
    # Split the request into an array of words
    words=($line)
    # Extract the HTTP method and path from the request
    method=${words[0]}
    path=${words[1]}

    # Only handle GET requests for now
    if [ "$method" = "GET" ]; then
      # Send a basic HTML response
      cat <<EOF | nc -l -p 8080 >&0
HTTP/1.1 200 OK
Content-Type: text/html

<html>
<head>
<title>Hello World!</title>
</head>
<body>
<h1>Hello World!</h1>
<p>This is a basic HTML response.</p>
</body>
</html>
EOF
    else
      # Send a 405 "method not allowed" error for all other request types
      cat <<EOF | nc -l -p 8080 >&0
HTTP/1.1 405 Method Not Allowed
Content-Type: text/plain

405 Method Not Allowed
EOF
    fi
  done
done

```

This script starts listening on port 8080 for incoming requests using netcat ("nc"). It maps standard in (stdin) and standard out (stdout) to file pipes to communicate with the nc process. The script then enters an infinite loop to handle incoming requests.

For each request, the script extracts the HTTP method and path from the request and checks if the method is "GET". If it is, the script sends a basic HTML response with a "200 OK" status code. If the method is anything other than "GET", the script sends a "405 Method Not Allowed" error response.

After each request is handled, the nc process is restarted to listen for the next incoming request.
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