devtoolset-8 was only released a short while ago. The linked installation instructions may be of use. However, your question pertains to CentOS, and this does not yet appear to have been made available yet. You can see some evidence of it being build for CentOS here, but it's not been updated for the final release yet.

You could ask on the SCL mailing list for an ETA, or wait until it appears in its final form. In the meantime, you could download the RPMs from koji directly.

According to that article, you can check if you have access to Red Hat Software Collections (RHSCL) by running the following command by the root user:

$ su -
# subscription-manager repos --list | egrep rhscl

If you have, enable necessary software repo and then install devtoolset:

# subscription-manager repos --enable rhel-7-server-optional-rpms
# yum install devtoolset-8

I ran into the same issue that you are facing and this is how I got it fixed:

1. Just want to be careful, you need to exist the Terminal and open a fresh one to start. This way, you are not under any devtoolset's bash.
2. Go to /opt/rh folder, run command ls -la to see if you have any devtoolset-* folder there. Let's say you have devtoolset-8, proceed step 2.
3. Go to /etc/scl/prefixes folder, if you don't see devtoolset-8 file, you can create a new one as devtoolset-8, and type 1 line: /opt/rh, then save and quit that file.
4. Once you are done, you can call: scl enable devtoolset-8 -- bash w/o any error. Good luck

An alternative of source /opt/rh/devtoolset-4/enable is

source scl_source enable devtoolset-4

The above shell script scl_source is more elegant than using a hard coded path (may be different on another machine). However scl_source does less because /opt/rh/devtoolset-4/enable uses scl_source and other stuff.

To use scl_source you may have to upgrade package scl-utils

yum update scl-utils  # old scl-utils versions miss scl_source

Quick copy-paste

echo 'source scl_source enable devtoolset-4' >> ~/.bashrc
    # Do not forget to change the version ↑

Source code for curious people

An example of scl_source source code:
https://gist.github.com/bkabrda/6435016

The scl_source installed on my Red Hat 7.1

#!/bin/bash

_scl_source_help="Usage: source scl_source <action> [<collection> ...]

Don't use this script outside of SCL scriptlets!

Options:
    -h, --help    display this help and exit"

if [ $# -eq 0 -o $1 = "-h" -o $1 = "--help" ]; then
    echo "$_scl_source_help"
    return 0
fi


if [ -z "$_recursion" ]; then
    _recursion="false"
fi
if [ -z "$_scl_scriptlet_name" ]; then
    # The only allowed action in the case of recursion is the same
    # as was the original
    _scl_scriptlet_name=$1
fi
shift 1

if [ -z "$_scl_dir" ]; then
    # No need to re-define the directory twice
    _scl_dir=/etc/scl/conf
    if [ ! -e $_scl_dir ]; then
        _scl_dir=/etc/scl/prefixes
    fi
fi

for arg in "$@"; do
    _scl_prefix_file=$_scl_dir/$arg
    _scl_prefix=`cat $_scl_prefix_file 2> /dev/null`
    if [ $? -ne 0 ]; then
        echo "Can't read $_scl_prefix_file, $arg is probably not installed."
        return 1
    fi

    # First check if the collection is already in the list
    # of collections to be enabled
    for scl in ${_scls[@]}; do
        if [ $arg == $scl ]; then
            continue 2
        fi
    done

    # Now check if the collection isn't already enabled
    /usr/bin/scl_enabled $arg > /dev/null 2> /dev/null
    if [ $? -ne 0 ]; then
        _scls+=($arg)
        _scl_prefixes+=($_scl_prefix)
    fi;
done

if [ $_recursion == "false" ]; then
    _i=0
    _recursion="true"
    while [ $_i -lt ${#_scls[@]} ]; do
        _scl_scriptlet_path="${_scl_prefixes[$_i]}/${_scls[$_i]}/${_scl_scriptlet_name}"
        source "$_scl_scriptlet_path"
        if [ $? -ne 0 ]; then
            echo "Can't source $_scl_scriptlet_name, skipping."
        else
            export X_SCLS="${_scls[$_i]} $X_SCLS"
        fi;
        _i=$(($_i+1))
    done
    _scls=()
    _scl_prefixes=()
    _scl_scriptlet_name=""
    _recursion="false"
fi