I just stumbled over this questions because I had a similar problem: The type of my Chart.js chart was 'horizontalBar' and for some datasets (where no zero values was present across the dataset) the horizontal bar did not start with 0 rather with the lowest value from the dataset.

I tried to figure out a solution and came up with following entry in the options object while creating the charts:

ticks: {
                  beginAtZero:true,
                  mirror:false,
                  suggestedMin: 0,
                  suggestedMax: 100
} 

However that did not work as expected although all posts said it works that way. After further investigation and reading of the Chart.js documentation I found the solution. The reason the further step did not work was following I found in the documentation:

However, any options specified on the x axis in a bar chart, are applied to the y axis in a horizontal bar chart.

So I just changed my options object to hold the proper configuration for the xAxes and it worked.

For those who are interested here is the the whole code I used for creating the horizontal bar chart with y-axis starting always at zero:

this.chart = new Chart(
    ctx,
    {
        type: 'horizontalBar',
        data: this.data.chartdata,
        options: {
          scales: {
              xAxes: [{ 
                stacked: false,
                ticks: {
                  beginAtZero:true,
                  mirror:false,
                  suggestedMin: 0,
                  suggestedMax: 100
                } 
              }],
              yAxes: [{ 
                stacked: true
              }]
          },
          scaleBeginAtZero : true,
          // important here to use () =>
          // to keep the scope of this
          onClick: (e) => {
            var actChart : Chart = this.charts[trialId];
            var element =
              actChart.getElementAtEvent(e);

            }
        }
    }
);

Here is the simplest way to do this in V3 chart js

Chart.defaults.datasets.bar.minBarLength = 5;

2019 Update

This can be done easily as below.

var myChart = new Chart(ctx, {
    ...
    options: {
        ...
        scales: {
            yAxes: [{
                ticks: {
                    beginAtZero: true
                }
            }]
        }
    }
);

You can find this in Chart.js documentation https://www.chartjs.org/docs/latest/

If you struggle with this, here's what I came up with. It is similar idea to Li Jinyao, but in addition, you would get click and hover events (tooltip) working for whole bar. I value is close to 0 but negative, the bar will show on negative side of x axis - you can easily get rid of it if that's not what you want to do.

const zeroCompensation = {
id: 'zeroCompensation',

beforeDatasetsDraw: function(chart) {
    const meta = chart.getDatasetMeta(0)
    forEach(meta.data, d => {
        const barHeight = d._view.base - d._view.y;
        if(Math.abs(barHeight) < minBarHeight /* I used value 5 */) {
            d._view.y = d._view.base - minBarHeight * (Math.sign(barHeight) || 1);
        }
    });
}};

and add it to plugins:

plugins: [zeroCompensation]

Keep in mind that this will work for values close to 0, not only 0. If you want it only for zeroes, you can change contents of if condition to:

chart.config.data.datasets[0].data[index] === 0

This is what Li Jinyao used in his answer. Hope that helps.
Edit: I wanted to highlight that this solution works regardless of values spread. Answer marked as solution will not work as intended if there are some high values in data set - 0.1 will render same as 0 in that case.

After dig into the plugin system, if you using Chart.js >=2.5, you can write a plugin to achieve it. Here is an example to draw a line when data is zero.

Here is my code:

const zeroCompensation = {
  renderZeroCompensation: function (chartInstance, d) {
      // get postion info from _view
      const view = d._view
      const context = chartInstance.chart.ctx

      // the view.x is the centeral point of the bar, so we need minus half width of the bar.
      const startX = view.x - view.width / 2
      // common canvas API, Check it out on MDN
      context.beginPath();
      // set line color, you can do more custom settings here.
      context.strokeStyle = '#aaaaaa';
      context.moveTo(startX, view.y);
      // draw the line!
      context.lineTo(startX + view.width, view.y);
      // bam！ you will see the lines.
      context.stroke();
  },

  afterDatasetsDraw: function (chart, easing) {
      // get data meta, we need the location info in _view property.
      const meta = chart.getDatasetMeta(0)
      // also you need get datasets to find which item is 0.
      const dataSet = chart.config.data.datasets[0].data
      meta.data.forEach((d, index) => {
          // for the item which value is 0, reander a line.
          if(dataSet[index] === 0) {
            this.renderZeroCompensation(chart, d)
          }
      })
  }
};

and here is how to add the plugin to Chart.js

var chart1 = new Chart(ctx, {
    plugins: [zeroCompensation]
});

The offcial document is not clear about their plugin API, you may console.log to find what you want.

Simply specify minBarLength in the dataset, with the minimum length in pixels the bars should have. See documentation.

Working Example:

var $chartCanvas = $('myCanvas');
var barChart = new Chart(myCanvas, {
    type: 'bar',
    data: {
        labels: ['Accepted Answer', 'Top rated answer', 'This Answer'],
        datasets:[{
            data: [0, 3, 10],
            minBarLength: 7,    // This is the important line!
        }]
    },
    options: {
        title: {
            display: true,
            text: 'helpfulness of answers to this questions'
        },
        legend: {
            display: false
        }
    }
});

<script src="https://cdn.jsdelivr.net/npm/chart.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<canvas id="myCanvas"></canvas>