While Devtoolset-3 and 4 are located on the Centos mirror site, yum will not be able to find them as they were not included in the repodata on the site. These were probably excluded because they are EOL (end-of-life), and have dependency bugs (Bug 1410152). Try using the older Devtoolset-3 packages here, which do not have the dependency issues:

https://copr.fedorainfracloud.org/coprs/rhscl/devtoolset-3/

If you really want the packages from the centos.org site, I went with wget to download all the RPMs, then recreated a local yum repo. However, you'll see the dependency issues and yum install devtoolset-3 will fail.

$ wget -r -p -e robots=off --directory-prefix="/mnt/local-devtoolset3-repo" --recursive --no-clobber --no-parent http://mirror.centos.org/centos/6/sclo/x86_64/rh/devtoolset-3/
$ sudo yum install createrepo
$ cd /mnt/local-devtoolset3-repo
$ createrepo --database .

Install it by:

sudo yum install centos-release-scl
sudo yum install devtoolset-4

The first command installs and enables Software Collections Repository on your CentOS machine. That repository provides the devtoolset package.

To install the full tools-set including gfortran on centos 7:

yum install centos-release-scl
yum install devtoolset-8
scl enable devtoolset-8 -- bash

enable the tools:

source /opt/rh/devtoolset-8/enable 

you may wish to put the command above in .bash_profile

ref: https://unix.stackexchange.com/questions/477360/centos-7-gcc-8-installation

In your console output it says: Maybe run: yum groups mark install (see man yum)—did you do this?

Try running the following commands:

yum groups mark install "Development Tools"
yum groups mark convert "Development Tools"
yum groupinstall "Development Tools"

Reference: RedHat Customer Portal discussion

Now devtools-8 is available and it's possible to use it by following commands:

yum install centos-release-scl
yum install devtoolset-8-gcc devtoolset-8-gcc-c++
scl enable devtoolset-8 -- bash

It's explained in a little more detail in this answer on StackOverflow: How to install GCC/G++ 8 on CentOS. And of course, if you need GFortran, you can install it in the same way - the package is devtoolset-8-gcc-gfortran.

What I have so far:

cat Dockerfile

FROM centos:7 AS env

RUN yum update -y
RUN yum install -y centos-release-scl
RUN yum install -y devtoolset-9

RUN echo "source /opt/rh/devtoolset-9/enable" >> /etc/bashrc
SHELL ["/bin/bash", "--login", "-c"]
RUN gcc --version

So you must:

1. Add the source stuff in a bashrc
   note: On Centos it's /etc/bashrc while on ubuntu it's /etc/bash.bashrc

2. Update the docker default shell to be bash AND to "load" the bashrc using --login

Output

docker build .
Sending build context to Docker daemon  4.096kB
Step 1/32 : FROM centos:7 AS env
 ---> 8652b9f0cb4c
Step 2/32 : RUN yum update -y
 ---> Using cache
 ---> a2bb269cd8dc
Step 3/32 : RUN yum install -y centos-release-scl
 ---> Using cache
 ---> 1184e26c71cf
Step 4/32 : RUN yum install -y devtoolset-9
 ---> Using cache
 ---> e678665d2a4e
Step 5/32 : RUN echo "source /opt/rh/devtoolset-9/enable" >> /etc/bashrc
 ---> Using cache
 ---> fe1745d4ca87
Step 6/32 : SHELL ["/bin/bash", "--login", "-c"]
 ---> Running in 2dd7955f4487
Removing intermediate container 2dd7955f4487
 ---> 3cf4835bf680
Step 7/32 : RUN gcc --version
 ---> Running in b5de3266d607
gcc (GCC) 9.3.1 20200408 (Red Hat 9.3.1-2)
Copyright (C) 2019 Free Software Foundation, Inc.
This is free software; see the source for copying conditions.  There is NO
warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
 ...

What won't work

Test 1

RUN scl enable devtoolset-9 bash
RUN gcc --version | head -1

each RUN is a new shell so the sub-bash is lost on the second line.

Test 2

RUN source /opt/rh/devtoolset-9/enable && gcc --version | head -1
RUN gcc --version | head -1

Here again the source is only for the first RUN shell command but will be lost...

Test 3

This may work but with potential unexpected behaviour

ENV PATH=/opt/rh/devtoolset-9/root/bin:$PATH
RUN gcc --version | head -1

here we only "fix" the PATH variable but if you look at the /opt/rh/devtoolset-9/enable script there is so more to do than only updating the PATH...