[Vuejs]-Two checkboxes, either one checked or the other, not both, how in vue.js?

You should use radio type inputs for that case, using the same “name” attribute for both:

  <input type="radio" id="week" value="week" name="jobPostingLength" v-model="jobPostingLength">
  <label for="week">1 Week</label>

  <input type="radio" id="month" value="month" name="jobPostingLength" v-model="jobPostingLength">
  <label for="month">1 Month</label>