Continue Jenkins pipeline past failed stage

This is now possible. Below is an example of a declarative pipeline, but catchError works for scripted pipelines as well.

pipeline {
    agent any
    stages {
        stage('1') {
            steps {
                sh 'exit 0'
            }
        }
        stage('2') {
            steps {
                catchError(buildResult: 'SUCCESS', stageResult: 'FAILURE') {
                    sh "exit 1"
                }
            }
        }
        stage('3') {
            steps {
                sh 'exit 0'
            }
        }
    }
}

In the example above, all stages will execute, the pipeline will be successful, but stage 2 will show as failed:

As you might have guessed, you can freely choose the buildResult and stageResult, in case you want it to be unstable or anything else. You can even fail the build and continue the execution of the pipeline.

Just make sure your Jenkins is up to date, since this is a fairly new feature.

EDIT: You need "Pipeline: Basic Steps" 2.16 (May 14, 2019)

I had the same concern. I was able to resolve it doing this.

Second stage will show in red and be marked as failed while the rest of the stages will keep running. You can set a flag and at the end of the stages check the flag and inform the status of the whole build.

node {

    def build_ok = true

    stage('one') {
        sh 'exit 0'
    }

    try{
        stage('two') {
            sh 'exit 1'   // failure
        }
    } catch(e) {
        build_ok = false
        echo e.toString()  
    }

    stage('three') {
        sh 'exit 0'
    }

    ....

    if(build_ok) {
        currentBuild.result = "SUCCESS"
    } else {
        currentBuild.result = "FAILURE"
    }
}

It depends whether you are using declarative pipeline syntax or scripted pipeline syntax.

declarative pipeline syntax:

pipeline {
    agent any
    stages {
        stage('one') {
            steps {
                sh 'exit 0'
            }
        }
        stage('two') {
            steps {
                sh 'exit 1'   // failure
            }
        }
    }
    post {
        always {
            sh 'exit 0'
        }
    }
}

Post-condition blocks contain steps the same as the steps section.

scripted pipeline syntax:

node {

    def build_ok = true

    stage('one') {
        sh 'exit 0'
    }

    try{
        stage('two') {
            sh 'exit 1'   // failure
        }
    } catch(e) {
        build_ok = false
        echo e.toString()  
    }

    stage('three') {
        sh 'exit 0'
    }

    if(build_ok) {
        currentBuild.result = "SUCCESS"
    } else {
        currentBuild.result = "FAILURE"
    }
}

This should work. However all boxes are red if even only one fails, but you can see boxes with error marked, so you will easily distinguish failed jobs.

def indexes = ['one', 'two', 'three']

node() {
    for (index in indexes) {
        catchError {
            stage(index) {
                println index
                sh '''echo "123"'''
            }
        }
    }
}